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Let $$\dot{x}=f(x)$$ be a system of autonomous ordinary differential equation in $\mathbb{R}^n$ defined by a vector field $f\colon \mathbb{R}^n\to \mathbb{R}^n$ . A set $A$ is said to be an attracting set[GH,P] if
- $A$ is closed and invariant,
- there exists an open neighborhood $U$ of $A$ such that all solution with initial solution in $U$ will eventually enter $A$ ($x(t)\to A$ ) as $t\to \infty$ .
Additionally, if $A$ contains a dense orbit then $A$ is said to be an attractor[GH,P].
Conversely, a set $R$ is said to be a repelling set[GH] if $R$ satisfy the condition 1. and 2. where $t\to \infty$ is replaced by $t\to -\infty$ . Similarly, if $R$ contains a dense orbit then $R$ is said to be a repellor[GH].
- GH
- GUCKENHEIMER, JOHN & HOLMES, PHILIP, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983.
- P
- PERKO, LAWRENCE, Differential Equations and Dynamical Systems, Springer, New York, 2001.
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"attractor" is owned by Daume.
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| Also defines: |
attracting set, repelling set, repellor |
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Cross-references: conversely, orbit, dense, contains, eventually, solution, neighborhood, open, invariant, closed, vector field, ordinary differential equation, autonomous
There are 5 references to this entry.
This is version 1 of attractor, born on 2005-05-20.
Object id is 7089, canonical name is Attractor.
Accessed 4751 times total.
Classification:
| AMS MSC: | 34C99 (Ordinary differential equations :: Qualitative theory :: Miscellaneous) |
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Pending Errata and Addenda
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